Number Theory and Advanced Cryptography 3' Quadratic Residues and Rabin Public key Cryptosystem

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Number Theory and Advanced Cryptography 3.
Quadratic Residues and Rabin Public key
Cryptosystem
Part I Introduction to Number Theory Part II
Advanced Cryptography

• Chih-Hung Wang
• Feb. 2008

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Quadratic Residues
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Quadratic Residues
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Quadratic Residuosity Problem (1)
Proof see page 188
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Quadratic Residuosity Problem (2)
Important!
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Quadratic Residuosity Problem (3)
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Legendre-Jacobi Symbols
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Jacobi Symbol Properties
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Algorithm for Computing Jacobi Symbol
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Notes of Jacobi Symbol

• Note that the Jacobi symbol is not defined for
  or even.
• Testing web-site
• http//mathworld.wolfram.com/JacobiSymbol.html
• http//www.math.fau.edu/Richman/jacobi.htm
• http//wwwmaths.anu.edu.au/DoM/thirdyear/MATH3301/
  jacobi.html

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Square Root Modulo Integer(1)

• Modulo prime Algorithm

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Square Root Modulo Integer(2)

• Modulo prime in General case

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Square Root Modulo Integer(3)

• Modulo composite

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Square Root Modulo Integer(4)

• Properties of modulo composite

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Example (1)
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Example (2)
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Factoring Problem

• For a large n with large prime factors, factoring
  is a hard problem, but not as hard as it used to
  be.
• Example factorize 48770428682337401 gt hard
  problem
• Easy problem
• Is 223092871 a factor of 48770428682337401?
• 1977 three inventors of RSA issue Mathematical
  Games
• 100 reward
• 1994 RSA-129 (428 bits) breaking

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Progress of Factorization (1)
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Progress of Factorization (2)
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Progress of Factorization (3)
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Blum Integers
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Properties of Blum Integers (1)
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Properties of Blum Integers (2)
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Rabin Encryption Scheme (1)
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Rabin Encryption Scheme (1)
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Example of Rabin
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Insecurity of Rabin
CPA (Chosen-plaintext attack) CCA
(Chosen-ciphertext attack)